When the paper was lit, the air inside the bottle got hotter and
expanded. The hot air had nowhere to go as the egg blocked the
entrance to the milk bottle. When the flame went out, the air
inside the bottle cooled down and contracted. This caused the
pressure inside the bottle to decrease, making it lower than the
air pressure outside the bottle. This higher pressure outside the
bottle pushed the egg into the bottle.

When you blew into the milk bottle this increased the pressure
inside the milk bottle so that it was greater than the air pressure
outside the bottle. This pushes the egg out of the bottle. Very
clever!